The Markovian Regime-Switching Risk Model with Constant Dividend Barrier under Absolute Ruin

ABSTRACT

In this paper, we consider the dividend payments prior to absolute ruin in a Markovian regime-switching risk process in which the rate for the Poisson claim arrivals and the distribution of the claim amounts are driven by an underlying Markov jump process. A system of integro-differential equations with boundary conditions satisfied by the moment-generating function, the*n* th moment of the discounted dividend payments prior to absolute ruin and the expected discounted penalty function, given the initial environment state, are derived. Then, the matrix form of systems of integro-differential equations satisfied by the discounted penalty function are presented. Finally, we obtain the integro-differential equations satisfied by the time to reach the dividend barrier.

In this paper, we consider the dividend payments prior to absolute ruin in a Markovian regime-switching risk process in which the rate for the Poisson claim arrivals and the distribution of the claim amounts are driven by an underlying Markov jump process. A system of integro-differential equations with boundary conditions satisfied by the moment-generating function, the

KEYWORDS

Absolute Ruin, Debit Interest, Moment-Generating Function, Markovian Regime-Switching Risk Model, Dividend Barrier, Integro-Differential Equation

Absolute Ruin, Debit Interest, Moment-Generating Function, Markovian Regime-Switching Risk Model, Dividend Barrier, Integro-Differential Equation

Cite this paper

nullW. Yu and Y. Huang, "The Markovian Regime-Switching Risk Model with Constant Dividend Barrier under Absolute Ruin,"*Journal of Mathematical Finance*, Vol. 1 No. 3, 2011, pp. 83-89. doi: 10.4236/jmf.2011.13011.

nullW. Yu and Y. Huang, "The Markovian Regime-Switching Risk Model with Constant Dividend Barrier under Absolute Ruin,"

References

[1] [1] J. M. Reinhard, “On a Class of Semi-Markov Risk Models Obtained as Classical Risk Models in a Markovian Environment,” ASTIN Bulletin, Vol. 14, 1984, pp. 23-43.

[2] S. Asmussen, “Risk Theory in a Markovian Environment,” Scandinavian Actuarial Journal, Vol. 2, 1989, pp. 69-100.

[3] Y. Lu and S. Li, “On the Probability of Ruin in a Markov-modulated Risk Model,” Insurance: Mathemat- ics and Economics, Vol. 37, No. 3, 2005, pp. 522-532. doi:10.1016/j.insmatheco.2005.05.006

[4] A. Ng and H. Yang, “Lundberg-Type Bounds for the Joint Distribution of Surplus Immediately before and after Ruin under a Markov-modulated Risk Model,” Astin Bulletin, Vol. 35, 2005, pp. 351-361. doi:10.2143/AST.35.2.2003457

[5] A. Ng and H. Yang, “On the Joint Distribution of Surplus Prior and Immediately after Ruin under a Markovian Re- gime Switching Model,” Stochastic Processes and Their Applications, Vol. 116, No. 2, 2006, pp. 244-266. doi:10.1016/j.spa.2005.09.008

[6] S. M. Li and Y. Lu, “Moments of the Dividend Payments and Related Problems in a Markov-Modulated Risk Mo- del,” North American Actuarial Journal, Vol. 11, No. 2, 2007, pp. 65-76.

[7] Y. Lu and S. Li, “The Markovian Regime-switching Risk Model with a Threshold Dividend Strategy,” Insurance: Mathematics and Economics, Vol. 44, No. 2, 2009, pp. 296-303. doi:10.1016/j.insmatheco.2008.04.004

[8] J. Liu, J. C. Xu and H. C. Hu, “The Markov-Dependent Risk Model with a Threshold Dividend Strategy,” Wuhan University Journal of Natural Sciences, Vol. 16, No. 3, 2011, pp. 193-198. doi:10.1007/s11859-011-0736-9

[9] J. Zhu and H. Yang, “Ruin Theory for a Markov Regime-Switching Model under a Threshold Dividend Stra- tegy,” Insurance: Mathematics and Economics, Vol. 42, No. 1, 2008, pp. 311-318. doi:10.1016/j.insmatheco.2007.03.004

[10] J. Q. Wei, H. L. Yang and R. M. Wang, “On the Markov -Modulated Insurance Risk Model with Tax,” Blaetter der DGVFM, Vol. 31, No. 1, 2010, pp. 65-78. doi:10.1007/s11857-010-0104-4

[11] M. Zhou and C. Zhang, “Absolute Ruin under Classical Risk Model,” Acta Mathematicae Applicate Sinica, Vol. 28, No. 4, 2005, pp. 57-80.

[12] J. Cai, “On the Time Value of Absolute Ruin with Debit Interest,” Advances in Applied Probability, Vol. 39, No. 2, 2007, pp. 343-359. doi:10.1239/aap/1183667614

[13] H. L. Yuan and Y. J. Hu, “Absolute Ruin in the Compound Poisson Risk Model with Constant Dividend Barrier,” Statistics and Probability Letter, Vol. 78, No. 14, 2008, pp. 2086-2094. doi:10.1016/j.spl.2008.01.076

[14] C. W. Wang and C. C. Yin, “Dividend Payments in the Classical Risk Model under Absolute Ruin with Debit In- terest,” Applied stochastic models in business and Indus- try, Vol. 25, No. 3, 2009, pp. 247-262. doi:10.1002/asmb.722

[15] C. W. Wang, C. C.Yin and E. Q. Li, “On the Classical Risk Model with Credit and Debit Interests under Absolute Ruin,” Statistics and Probability Letters, Vol. 80, No. 15, 2010, pp. 427-436. doi:10.1016/j.spl.2009.11.020

[1] [1] J. M. Reinhard, “On a Class of Semi-Markov Risk Models Obtained as Classical Risk Models in a Markovian Environment,” ASTIN Bulletin, Vol. 14, 1984, pp. 23-43.

[2] S. Asmussen, “Risk Theory in a Markovian Environment,” Scandinavian Actuarial Journal, Vol. 2, 1989, pp. 69-100.

[3] Y. Lu and S. Li, “On the Probability of Ruin in a Markov-modulated Risk Model,” Insurance: Mathemat- ics and Economics, Vol. 37, No. 3, 2005, pp. 522-532. doi:10.1016/j.insmatheco.2005.05.006

[4] A. Ng and H. Yang, “Lundberg-Type Bounds for the Joint Distribution of Surplus Immediately before and after Ruin under a Markov-modulated Risk Model,” Astin Bulletin, Vol. 35, 2005, pp. 351-361. doi:10.2143/AST.35.2.2003457

[5] A. Ng and H. Yang, “On the Joint Distribution of Surplus Prior and Immediately after Ruin under a Markovian Re- gime Switching Model,” Stochastic Processes and Their Applications, Vol. 116, No. 2, 2006, pp. 244-266. doi:10.1016/j.spa.2005.09.008

[6] S. M. Li and Y. Lu, “Moments of the Dividend Payments and Related Problems in a Markov-Modulated Risk Mo- del,” North American Actuarial Journal, Vol. 11, No. 2, 2007, pp. 65-76.

[7] Y. Lu and S. Li, “The Markovian Regime-switching Risk Model with a Threshold Dividend Strategy,” Insurance: Mathematics and Economics, Vol. 44, No. 2, 2009, pp. 296-303. doi:10.1016/j.insmatheco.2008.04.004

[8] J. Liu, J. C. Xu and H. C. Hu, “The Markov-Dependent Risk Model with a Threshold Dividend Strategy,” Wuhan University Journal of Natural Sciences, Vol. 16, No. 3, 2011, pp. 193-198. doi:10.1007/s11859-011-0736-9

[9] J. Zhu and H. Yang, “Ruin Theory for a Markov Regime-Switching Model under a Threshold Dividend Stra- tegy,” Insurance: Mathematics and Economics, Vol. 42, No. 1, 2008, pp. 311-318. doi:10.1016/j.insmatheco.2007.03.004

[10] J. Q. Wei, H. L. Yang and R. M. Wang, “On the Markov -Modulated Insurance Risk Model with Tax,” Blaetter der DGVFM, Vol. 31, No. 1, 2010, pp. 65-78. doi:10.1007/s11857-010-0104-4

[11] M. Zhou and C. Zhang, “Absolute Ruin under Classical Risk Model,” Acta Mathematicae Applicate Sinica, Vol. 28, No. 4, 2005, pp. 57-80.

[12] J. Cai, “On the Time Value of Absolute Ruin with Debit Interest,” Advances in Applied Probability, Vol. 39, No. 2, 2007, pp. 343-359. doi:10.1239/aap/1183667614

[13] H. L. Yuan and Y. J. Hu, “Absolute Ruin in the Compound Poisson Risk Model with Constant Dividend Barrier,” Statistics and Probability Letter, Vol. 78, No. 14, 2008, pp. 2086-2094. doi:10.1016/j.spl.2008.01.076

[14] C. W. Wang and C. C. Yin, “Dividend Payments in the Classical Risk Model under Absolute Ruin with Debit In- terest,” Applied stochastic models in business and Indus- try, Vol. 25, No. 3, 2009, pp. 247-262. doi:10.1002/asmb.722

[15] C. W. Wang, C. C.Yin and E. Q. Li, “On the Classical Risk Model with Credit and Debit Interests under Absolute Ruin,” Statistics and Probability Letters, Vol. 80, No. 15, 2010, pp. 427-436. doi:10.1016/j.spl.2009.11.020